Let us write the two complex numbers in polar coordinates and let them be
#z_1=r_1(cosalpha+isinalpha)# and #z_2=r_2(cosbeta+isinbeta)#
Here, if two complex numbers are #a_1+ib_1# and #a_2+ib_2# #r_1=sqrt(a_1^2+b_1^2)#, #r_2=sqrt(a_2^2+b_2^2)# and #alpha=tan^(-1)(b_1/a_1)#, #beta=tan^(-1)(b_2/a_2)#
Their division leads us to
#{r_1/r_2}{(cosalpha+isinalpha)/(cosbeta+isinbeta)}# or
#{r_1/r_2}{(cosalpha+isinalpha)/(cosbeta+isinbeta)xx(cosbeta-isinbeta)/(cosbeta-isinbeta)}#
#(r_1/r_2){(cosalphacosbeta+sinalphasinbeta)+i(sinalphacosbeta-cosalphasinbeta))/((cos^2beta+sin^2beta))# or
#(r_1/r_2)*(cos(alpha-beta)+isin(alpha-beta))# or
#z_1/z_2# is given by #(r_1/r_2, (alpha-beta))#
So for division complex number #z_1# by #z_2# , take new angle as #(alpha-beta)# and modulus the ratio #r_1/r_2# of the modulus of two numbers.
Here #2-4i# can be written as #r_1(cosalpha+isinalpha)# where #r_1=sqrt(2^2+(-4)^2)=sqrt20# and #alpha=tan^(-1)(-4/2)=tan^(-1)(-2)#
and #5+8i# can be written as #r_2(cosbeta+isinbeta)# where #r_2=sqrt(5^2+8^2)=sqrt89# and #beta=tan^(-1)(8/5)#
and #z_1/z_2=sqrt20/(sqrt89)(costheta+isintheta)#, where #theta=alpha-beta#
Hence, #tantheta=tan(alpha-beta)=(tanalpha-tanbeta)/(1+tanalphatanbeta)=((-2)-(8/5))/(1+(-2)xx(8/5))=(-18/5)/(-11/5)=18/11#.
Hence, #(2-4i)/(5+8i)=sqrt(20/89)(costheta+isintheta)#, where #theta=tan^(-1)(18/11)#
